Ensuring maths conceptual development through educationally sound integration and

Central to the NCS and key for the maths subject and the resulting primary maths teacher identity was the intended achievement of an optimal relationship between integration and conceptual progression which would ensure the acquisition of conceptual knowledge, the gaining/development of skills and the engendering of the necessary values and attitudes (DOE, 2002a; DOE, 2002b; DOE, 2002c; DOE, 2003). The NCS thus required a primary maths teacher who would dually ensure maths conceptual development in learners and look for necessary and educationally sound integration and learner-centred opportunities (DOE, 2002c; DOE, 2003). Such a primary maths teacher identity resonates with Bernstein (2000) and Tyler’s (1999) market pedagogic identity, which is orientated both towards the intrinsic value of the subject - typical of strong classification and a radically transformed regulative discourse (rules of the social order) of the weak frame.

2.5.1.1 Developing maths conceptual understanding and progression

The Learning Outcomes and their Assessment Standards19, which are described as “cognitively dependent and supportive of each other”, were the two critical curriculum design features that ensured maths conceptual progression and specified the depth and breadth of essential high level maths knowledge, skills and values to be covered and achieved by learners in each grade (DOE, 2002a; DOE, 2002c). In primary maths education the five key learning outcomes in the renamed ‘Mathematics’ (previously MLMMS) learning area statement were Numbers, operations and relationships; Patterns, functions and algebra; Space and shape; Measurement and Data Handling (DOE, 2002a; DOE, 2002b; DOE, 2002c;

18 _{The NCS adopted an ordered form of integration which according to Bernstein (1971, p. 222) has a “sense of} time, place and purpose”.

19 _{Learning outcomes describe what learners should know and be able to do. Assessment standards describe the} minimum level, depth and breadth of what is to be learnt (DOE, 2002a).

DOE, 2003). The first learning outcome was allocated 55% and 40% of the Foundation and Intermediate phase maths time respectively so as to ensure that learners develop a strong number sense, numerical knowledge, memorise multiplication fluently, sharpen mental calculation skills and perform efficient and accurate algorithms and solutions (DOE, 2002b; DOE, 2002c; DOE, 2003). The complementary third and fourth learning outcomes were allocated 30% of the primary maths learning time and content weighting because “measurement is a rich context for the development of Numbers, Operations and Relationships and Space and Shape a context for developing the early algebra skills of pattern recognition” (DOE, 2003, p. 21). The new curricular also had an increased focus on data and data handling so as to develop sensitivity to the power of data and to enable learners to learn the skills of collecting, summarising, displaying and critically analysing information (DOE, 2003; DOE, 2002c). In addition to the knowledge and skills listed in the Learning Outcomes the maths learning area also promoted problem solving, mathematical reasoning and offered opportunities for learners to communicate about mathematics, across the learning outcomes (DOE, 2003). The key learning outcomes and the promoted skills across the learning outcomes were meant to ensure that learners engage in worthwhile and challenging mathematical tasks that give opportunities for learners to develop a deep and coherent conceptual understanding of maths (DOE, 2003). Analysing these local primary maths knowledge’s orientations through Bernstein’s work points to a market pedagogic identity which is oriented towards the institutional discourse of the school and its intrinsic value (Bernstein, 2000). Thus the NCS focused on the key yet critical and basic concepts of school mathematics knowledge. Such a knowledge orientation under the NCS resulted, according to Muller (2006), in mathematics to be strongly classified.

2.5.1.2 Balancing integration and conceptual progression

The “achievement of an optimal relationship/balance between integration and conceptual progression” were central to this curriculum (DOE, 2002a, p. 13; DOE, 2003, p. 6; DOE, 2002c). However “conceptual development was not to be compromised by integration” (DOE, 2003, p. 26). I now turn to discuss how conceptual progression was embedded within teacher developed Learning Programmes. Teachers were ‘supported’ to develop their own Learning programmes through tightly sequenced and ordered guidelines (DOE, 2003). The principles and values of the NCS learning area statements underpinned these phase-long plans, which are defined as structured and systematic arrangements of activities that specify the scope of learning and promote the attainment of learning outcomes and assessment standards for the phase (DOE, 2002a; DOE, 2002c; DOE, 2003). The Foundation phase Numeracy and the Intermediate phase Mathematics learning programmes ensured that the

prescribed outcomes, the core knowledge, context and concept choices for this learning area are covered effectively and comprehensively in a sequential way across the phase, resulting in coherent, meaningful and relevant teaching, learning and assessment activities (DOE, 2002c; DOE, 2002a; DOE, 2003). The Learning programmes also contained work schedules of a yearlong programme on how teaching, learning and assessment was to be sequenced and paced in a particular grade (DOE, 2002a; DOE, 2003). The next level of planning involved a Lesson Plan which drew directly from the work schedule and provided detailed structure for teaching, learning and assessment activities (DOE, 2003). Overall the Learning Programmes, work schedules and lessons plans under the NCS took heed of the fact that “the learning of mathematics is developmental, hierarchical and dependent” – thus learners were supposed to be familiar with certain basic concepts before they dealt with more advanced mathematical concepts (DOE, 2003, p. 20). Under the NCS the primary maths learning area’s learning programmes, work schedules and lessons plans embedded (implicit) rules of selection, sequencing and pacing. According to Bernstein (2003) this is typical of a market- oriented visible pedagogy which is similar to a market pedagogic identity (Tyler, 1999). Strong rules of the discursive order can be read as part of measures to strengthen the frame of educational knowledge, especially the instructional discourse frame (Bernstein, 1971; Bernstein, 2000). The resulting primary maths teacher had to provide opportunities for learners to develop a deep and coherent conceptual understanding of maths and also enable structured conceptual progression.

2.5.1.3 Introducing Learning programmes and Learning support materials

Also to support conceptual links, progression and the selection, sequencing and pacing of key primary maths knowledge and skills was the introduction in the NCS of Learning Programmes/Learning areas time allocations and learning support materials which included manipulatives, textbooks, worksheets and technological devices (e.g. calculators, computers) (DOE, 2002a; DOE, 2003). The primary maths subject guidelines explained that textbooks were to be used coherently and in an orderly manner to support conceptual development in learners (DOE, 2003). Textbooks would also be relevant especially for the intermediate phase learners who were expected to work mathematical tasks independently and unsupervised (DOE, 2003). Bernstein (2003) argues that the textbook is highly prioritised in a market- visible pedagogy to enable learners to carry independent solitary work and to access alternative knowledge perspectives. In the Foundation phase the Numeracy learning programme was allocated 40% and the Intermediate phase Mathematics Learning area was given 18% of the total teaching time (a 15% and a 3% time increase respectively from C2005) (DOE, 2002a; DOE, 1997a; 1997b). Unlike C2005, which had time allocated and

reserved for flexibility and enrichment, the NCS had all its formal teaching time allocated for subject learning, and this, alongside strong pacing, indicates that time under this curriculum was at a premium (DOE, 1997a; DOE, 1997b; DOE, 2002a; Bernstein, 2003). The increase in numeracy and mathematics teaching time was also meant to indicate the strong emphasis and prioritising of this learning area, of course alongside literacy/languages. The increased time devoted for primary maths also indicates the status and the significance of this learning area/programme under the NCS (Bernstein, 1971).

2.5.1.4 Encouraging learner-centred and activity-based approaches

Like C2005, the Constitution of South Africa informed the revision of the NCS, which still “held dear the principles and practices of social justice, equity and democracy” and interweaved these goals across the curriculum (DOE, 2002a, p. 1 & 8; DOE, 2002c; DOE, 2003). Because the curriculum aimed at giving the Constitution practical expression in the classrooms, Outcomes-based education philosophy alongside the critical and developmental outcomes (inspired by the constitution) formed the foundation of the NCS (DOE, 2002a; DOE, 2002b; DOE, 2003). As an indication of embodying social democratic values in the school systems’ knowledge and skills, the NCS’ outcomes approach encouraged and emphasised a “participatory, learner-centred and activity-based approach to education” (DOE, 2002c, p. 1; DOE, 2002a, p. 12). Thus the curriculum, through the critical outcomes, expected learners to work effectively as members of a group or team, organise and manage themselves and make decisions using critical and creative thinking (DOE, 2002a; DOE, 2002c; DOE, 2003). The mathematics curriculum subject guidelines also gave room for primary teachers to innovatively and creatively determine appropriate teaching approaches (such as drill and practice, problem solving and investigation) that engage learners in worthwhile and challenging mathematical tasks, and gave them structured learning opportunities to develop a deep, coherent and interrelated understanding of mathematics (DOE, 2002a; DOE, 2002c; DOE, 2003). Teachers were also encouraged to provide learning opportunities that support “the different learning styles of the learners” taking heed of the fact that “learners learn at a different pace to each other” (DOE, 2003, p. 32 & 12). Teachers were also encouraged, where appropriate, to use the learners’ mother tongues and allow code switching in maths classes (DOE, 2003). Such personalised teaching practices are explained by Tyler (1999) as being characteristic of the market-pedagogic identity position. The primary maths curriculum guidelines thus encouraged the use of personalised practices, learner-centred approaches and teacher-oriented strategies that were discipline-focused and enhanced the conceptual understanding of maths.

2.5.1.5 Integration

The NCS also retained the principle of integration which is central and integral to outcomes- based education (DOE, 2002a; DOE, 2003). However the NCS had a different form of integration, compared to C2005’s more radical form of integration, which subordinated the learning of mathematics in pursuit of theme-based activity and learners’ personal everyday experiences (DOE, 1997a; DOE, 1997b; Ensor & Galant, 2005; Taylor & Vinjevold, 1999; Chisholm et al, 2000; Graven, 2002a; Reeves & Muller, 2005; Fleisch, 2008). Under the NCS, integration was driven by ‘learning outcomes’ rather than ‘themes’, and it was also not only limited to linking and relating the eight Learning Areas but meant to support “conceptual development rather than being introduced for its own sake” (DOE, 2003, p. 6 & 26). Integration and interrelations of mathematical concepts both within and across the learning outcomes was encouraged; with relevant social, economic, cultural and political contexts at the local, national and international level being selected to enhance the understanding of mathematical skills and knowledge. Furthermore teacher collaboration team planning approach for work schedules also promoted integration. With regard to integration, the NCS adopted an ordered form of integration which, according to Bernstein (1971, p. 222), produces a learning and teaching culture that has a “sense of time, place and purpose”.